\magnification = 2200 %\magstep3

\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input OurATOMacros
\input OurPlainGraphicsMacros

\hsize 7 true in 
\vsize 9 true in
\hoffset = -0.20 true in 
\voffset -0.25 true in
\parskip=3pt

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum

\nopagenumbers
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\cl{{\bf The Nephroid }
 \footnote{*}{\verysmall 
 This file is from the 3D-XplorMath project.  Please see: \hfill\break
   \phantom{http://} http://3D-XplorMath.org/}}

%\vskip4pt
\lf
The Nephroid is generated by rolling a circle of one radius on the 
outside of a second circle of twice the radius. In 3D-XplorMath, 
either choose {\tt Nephroid} from the Plane Curve menu, or
choose {\tt Circle},  then select {\tt Set Parameters...} 
from the Settings menu and set $hh = - 0.5\cdot aa$, 
$ii = 1$. With $R = 3r$ we thus have the parametrization for Nephroids:

\noindent
\hbox{ 
\vbox{\hsize=0.4\hsize \phantom{.}\hskip-8mm
\includegraphics[width=1.2in] {Nephroid.png}}
\vbox{\hsize=0.6\hsize \noindent
$$\eqalign{     
  x(t)  =&  R\cdot \sin(t) +  r\cdot \sin(3t)   \cr 
  y(t)  =&  R\cdot \cos(t) + r\cdot \cos(3t)  \cr
  }$$ \vskip1cm}}
 
 \noindent
As with Cardioids and Lima\c cons, one can also make the radius for the
drawing stick shorter or longer: Set $ii > 1$ for the looping relatives
of the Nephroid or see the default {\tt Morph} in the Animation menu.

\noindent
The complex map $z \mapsto z^3+3z $ maps the unit circle to such a
Nephroid. To see this, in the Conformal Map Category, select 
$z \mapsto z^ee + ee z$ 
 from the Conformal Map menu, and then choose Set Parameters from
the Settings menu and set $ee$ to $3$.

\noindent
The normals of one Nephroid have as envelope another, smaller
Nephroid---the same  phenomenon as for the Cardioid and the Cycloid. (To
see this select  {\tt Show Osculating Circles With Normals} from the Action
menu). In technical jargon, the caustics for each of these curves is a
similar  curve.

\noindent
H.K.

\bye